2017/01/03 by Yuan Xu, Xu, Yuan
Mathematics · #33C45 #33C50 #42A82 #60E10 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical functions and polynomials #Numerical methods in inverse problems #Point processes and geometric inequalities
paper · pdf · doi:10.48550/arxiv.1701.00787
openalex publication_date 2017/01/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
It is shown that the integrals of the Jacobi polynomials % ∫0t (t-θ)δPn(α-\frac12,β-\frac12)(cos θ) (sin \tfracθ2)2 α (cos \tfracθ2)2 β dθgt; 0 for all t ∈ (0,π] and n ∈ ℕ if δ≥ α+ 1 for α,β∈ ℕ0 and max\α,β\ > 0. This proves a conjecture on the integral of the Gegenbauer polynomials in \citeBCX that implies the strictly positive definiteness of the function θ↦ (t - θ)+δ on the unit sphere \mathbbSd-1 for δ≥ \lceil (d)/(2)\rceil and the Polyà criterion for positive definite functions on the sphere for all dimensions. Moreover, the positive definiteness of the function θ↦ (t - θ)+δ is also established on the compact two-point homogeneous spaces.